Given a discrete system, physical operators supported on the subsystem \(B\)  can be described by a local algebra, \(\mathcal {A}_B\) , which is the operator algebra An operator algebra is a linear space of operators (including the identity) which is closed under operator multiplication and Hermitian conjugation. Mathematically, it's a von Neumann algebra. of all the physical operators acting on \(B\) . By "physical operator", I mean it's gauge-invariant and bonsonic Operator algebra can also include fermionic operators. In this case, the algebra is a \(Z_2\) -graded algebra, which is different from the definition of local algebras in some literature, e.g. [1]}, where only bosonic operators are included.. If we can define the measure For example, in the case of qubit model, \(|B|\)  can be the number of qubits in \(B\) . Since we are considering discrete system, we can suppose \(|B|\in \mathbb {Z}\)  without loss of generality. of \(B\) , denoted as \(|B|\) , then we can attribute it to \(\mathcal {A}_B\)  as a label. Since \(\mathcal {A}_B\)  is closed under addition and scalar multiplication, \(\mathcal {A}_B\)  is also a linear space. When we hope to emphasize this fact, we denote \(\mathcal {A}_B\)  as \(A_B\) . Then we define the linear space \(A^k\)  as
\(A^k\equiv \sum _{|B|=k}A_B,\) 
